Concept Development Lessons - Mathshell
Concept Development LessonsHow can I help students develop a deeper understanding of Mathematics?A PROFESSIONAL DEVELOPMENT MODULEIntroductionThe Formative Assessment Lessons are of two types; those that focus on the development of conceptualunderstanding and those that focus on problem solving. Concept Development lessons are intended toassess and develop students’ understanding of fundamental concepts through activities that engage themin classifying and defining, representing concepts in multiple ways, testing and challenging commonmisconceptions, and exploring structure. Problem Solving lessons are intended to assess and developstudents’ capacity to select and deploy their mathematical knowledge in non-routine contexts andtypically involve students in comparing and critiquing alternative approaches to solving a problem.In this PD module, we focus on Concept Development lessons. Research has shown that individual,routine practice on standard problems does little to help students deepen their understanding ofmathematical concepts. Teaching becomes more effective when existing interpretations (andmisinterpretations) of concepts are shared and systematically explored within the classroom. The lessonsdescribed here typically begin with a formative assessment task that exposes students’ existing ways ofthinking. The teacher is then offered specific suggestions on how these may be challenged and developedthrough collaborative activities. New ideas are constructed through reflective discussion. This processplaces considerable pedagogic demands on teachers, and it is these demands that this module is intendedto explore.ActivitiesActivity A: Using the assessment tasks . 2Activity B: What causes mistakes and misconceptions?. 4Activity C: The Formative assessment lesson. . 6Activity D: Classifying mathematical objects . 8Activity E: Interpreting multiple representations . 10Activity F: Evaluating mathematical statements . 13Activity G: Exploring the structure of problems . 15Activity H: Plan a lesson, teach it and reflect on the outcomes . 17MAP Lessons for Formative Assessment of Concept Development . 18Acknowledgement:Parts of this material, including the video, were adapted from Improving Learning in Mathematics, a government funded programin the UK. See: Swan, M; (2005). Improving Learning in Mathematics, challenges and strategies, Department for Education andSkills Standards Unit. Obtainable in the UK from http://tlp.excellencegateway.org.uk/pdf/Improving learning in maths.pdfDraft Feb 2012 2012 MARS, Shell Centre, University of Nottingham
Activity A: Using the assessment tasksTime needed: 30 minutes.Each Formative Assessment Lesson is preceded by an introductory assessment task. The purpose of this isto discover the interpretations and understandings that students bring to this particular area ofmathematical content. This task is given to individual students a day or more before the main lesson andthe information gathered from student responses are then used to plan and direct the lesson.In this activity, participants begin to look at a selection of such assessment tasks and consider the kind ofinformation they provide, and how best to respond to students.The examples used below are taken from the following lessons: Interpreting distance-time graphs (Middle School) Increasing and decreasing quantities by a percent (Middle School) Interpreting algebraic expressions (High School)It would be helpful if participants could see the complete materials1 for one of these.Look at the assessment tasks from three lessons on Handout 1. Try to anticipate the kinds of mistakes your students would make on each of these tasks.What common difficulties would you expect?Now look at the samples of student work on Handout 2. What does the student appear to understand? Where is your evidence?List the errors and difficulties that are revealed by each response.Try to identify the thinking that lies behind each error.What feedback would you give to each student? Write down your comments on the work.Following each assessment task, we have provided suggestions for follow-up questions that wouldmove students’ thinking forward. These are given on Handout 3. Compare the feedback you havewritten to these questions. Do you normally give feedback to students in the form of questions?What are the advantages of using questions rather than more directive guidance?Can you suggest better questions to the ones provided?Research has shown that giving students scores or grades on their work is counter-productive, and thisshould not be done with these assessment tasks. This is discussed in more detail in ProfessionalDevelopment Module 1, ‘Formative Assessment.’1 Available from D Module GuideConcept Development Lessons2
Handout 1: Assessment tasksInterpreting Distance–Time GraphsStudent MaterialsHandout 2: Sample student workBeta VersionJourney to the Bus StopEvery morning Tom walks along a straight road from his home to a bus stop, a distance of 160 meters.The graph shows his journey on one particular day.1. Describe what may have happened.You should include details like how fast he walked.2. Are all sections of the graph realistic? Fully explain your answer. 2011 MARS University of Nottingham3 Sample follow-up questionsS-1Interpreting Distance-TimeGraphsTeacher GuideDistance-timegraphs:CommonHandout3: Samplefollow-upissuesquestionsCommon issues:Student interprets the graph as a pictureFor example: The student assumes that as the graphgoes up and down, Tom's path is going up anddown.Or: The student assumes that a straight line on agraph means that the motion is along a straight path.Or: The student thinks the negative slope meansTom has taken a detour.Student interprets graph as speed–timeThe student has interpreted a positive slope asspeeding up and a negative slope as slowing down.Beta VersionSuggested questions and prompts: If a person walked in a circle around theirhome, what would the graph look like? If a person walked at a steady speed up anddown a hill, directly away from home, whatwould the graph look like? In each section of his journey, is Tom's speedsteady or is it changing? How do you know? How can you figure out Tom's speed in eachsection of the journey? If a person walked for a mile at a steady speed,away from home, then turned round and walkedback home at the same steady speed, whatwould the graph look like? How does the distance change during thesecond section of Tom's journey? What doesthis mean? How does the distance change during the lastsection of Tom's journey? What does thismean? How can you tell if Tom is traveling away fromor towards home? Can you provide more information about howStudent fails to mention distance or timefar Tom has traveled during different sectionsFor example: The student has not mentioned howof his journey?far away from home Tom has traveled at the end of Canyou provide more information about howPD ModuleGuideConcept DevelopmentLessonseach section.much time Tom takes during different sectionsOr: The student has not mentioned the time for eachof his journey?section of the journey.3
Activity B: What causes mistakes and misconceptions?Time needed: 15 minutes.This activity is intended to encourage teachers to see that student errors may be due to deep-rootedmisconceptions that should be exposed and discussed in classrooms. Why do students make mistakes in Mathematics?What different types of mistakes are there? What are their causes?How do you respond to each different type? Why?Draw out the different possible causes of mistakes. These may be due to due to lapses in concentration,hasty reasoning, memory overload or a failure to notice important features of a problem. Other mistakes,however, may be symptoms of alternative ways of reasoning. Such ‘misconceptions’ should not bedismissed as ‘wrong thinking’ as they may be necessary stages of conceptual development.Consider generalizations commonly made by students, shown on Handout 4. Can you contribute some more examples to this list? Can you think of any misconceptions you have had at some time? How were these overcome?Many ‘misconceptions’ are the results of students making generalizations from limited domains. Forexample, when younger children deal solely with natural numbers they infer that ‘when you multiply byten you just add a zero.’ Later on, this leads to errors such as 3.4 x10 3.40. For what domains do the following generalizations work? When do they become invalid?o If I subtract something from 12, the answer will be smaller than 12.o The square root of a number is smaller than the number.o All numbers may be written as proper or improper fractions.o The order in which you multiply does not matter. Can you think of other generalizations that are only true for limited domains? There are two common ways of reacting to pupils’ errors and misconceptions:Avoid them whenever possible: “If I warn pupils about the misconceptions as I teach, they areless likely to happen. Prevention is better than cure.”Use them as learning opportunities: “I actively encourage learners to make mistakes and learnfrom them.”What are your views? Discuss the principles given on Handout 5. This describes the advice given in the research.How do participants feel about this advice?PD Module GuideConcept Development Lessons4
Handout 4: Generalizations commonly made by studentsHandout 5: Principles to discussPD Module GuideConcept Development Lessons5
Activity C: The Formative assessment lesson.Time needed: 20 minutes.This activity is designed to help participants recognize some of the broad principles upon which theFormative Assessment Lessons are designed.Choose one of the Formative assessment lessons that relates to the assessment tasks you looked at inActivity 1:o Interpreting distance-time graphso Increasing and decreasing quantities by a percento Interpreting algebraic expressions Compare the lesson with the broad structure described on Handout 6.How far does the lesson follow this structure and where does it deviate?Can you see reasons why the lessons have been designed in this way?Try to describe these reasons.Which type of activity has been used in the main activity? (See Handout 7)Work on the main activity together.How does the main activity address the common conceptual issues that you considered in thestudent work on Handout 2?How are the students expected to learn from the activity?On the following pages, we further describe and illustrate each of the four task genres shown on Handout7.PD Module GuideConcept Development Lessons6
Handout 6: Structure of the Conceptlessons5 Structure of the Concept lessonsBroadly speaking, each Concept Formative Assessment Lesson is structured in the followingway, with some variation, depending on the topic and task: (Before the lesson) Students complete an assessment task individuallyThis assessment task is designed to clarify students’ existing understandings of theconcepts under study. The teacher assesses a sample of these and plans appropriatequestions that will move student thinking forward. These questions are then introducedin the lesson at appropriate points. Whole class introductionEach lesson begins with the teacher presenting a problem for class discussion. Theaim here is to intrigue students, provoke discussion and/or model reasoning, Collaborative work on a substantial activityAt this point the main activity is introduced. This activity is designed to be a rich,collaborative learning experience. It is both accessible and challenging; having multipleentry points and multiple solution paths. It is usually done with shared resources and ispresented on a poster.Four types of activity are commonly used as shown in Handout 6. Students areinvolved in:ooooclassifying mathematical objects & challenging definitionsinterpreting multiple representationsevaluating conjectures and assertionsmodifying situations & exploring their structureThese will be explored more fully later in this module. It is not necessary for everystudent to complete the activity. Rather we hope that students will come to understandthe concepts more clearly. Students share their thinking with the whole classStudents now share some of their learning with other students. It is through explainingthat students begin to clarify their own thinking. The teacher may then ask furtherquestions to provoke deeper reflection. Students revisit the assessment taskFinally, students are asked to look again at their original answers to the assessmenttask. They are either asked to improve their responses or are asked to complete asimilar task. This helps both the teacher and the student to realize what has beenlearned from the lesson.Handout 7: Genres of activity used in the Concept lessonsDraft 2011University ofNottingham6 Some genresofMARSactivityusedin the Concept Lessons11The main activities in the concept lessons are built around the following four genres. Each ofthese types of activity is designed to provoke students to reason in different ways; torecognize properties, to define, to represent, to challenge conjectures and misconceptions, torecognize deeper structures in problems.1. Classifying mathematical objectsMathematics is full of conceptual ‘objects’ such as numbers, shapes, and functions. Inthis type of activity, students examine objects carefully, and classify them according totheir different attributes. Students have to select an object, discriminate between thatobject and other similar objects (what is the same and what is different?) and createand use categories to build definitions. This type of activity is therefore powerful inhelping students understand different mathematical terms and symbols, and theprocess by which they are developed.2. Interpreting multiple representationsMathematical concepts have many representations; words, diagrams, algebraicsymbols, tables, graphs and so forth. These activities allow different representations tobe shared, interpreted, compared and grouped in ways that allow students to constructmeanings and links between the underlying concepts.3. Evaluating mathematical statementsThese activities offer students a number of mathematical statements orgeneralizations. These statements may typically arise from student misconceptions, forexample: “The square root of a number is smaller than the number”. Students areasked to decide on their validity and give explanations for their decisions. Explanationsusually involve generating examples and counterexamples to support or refute thestatements. In addition, students may be invited to add conditions or otherwise revisethe statements so that they become ‘always true’.4. Exploring the structure of problemsIn this type of activity, students are given the task of devising their own mathematicalproblems. They try to devise problems that are both challenging and that they knowthey can solve correctly. Students first solve their own problems and then challengeother students to solve them. During this process, they offer support and act as‘teachers’ when the problem solver becomes stuck. Creating and solving problemsmay also be used to illustrate doing and undoing processes in mathematics. Forexample, one student might draw a circle and calculate its area. This student is thenasked to pass the result to a neighbor, who must now try to reconstruct the circle fromthe given area. Both students then collaborate to see where mistakes have arisen.PD Module GuideDraft 2011 MARS University of NottinghamConcept Development Lessons127
Activity D: Classifying mathematical objectsTime needed: 20 minutes.Mathematically proficient students try to communicate precisely to others. They try to use clear definitionsin discussion with others and in their own reasoning. (Common Core State Standards, p.7)Understanding a concept involves four mental processes: bringing it to the foreground of attention,naming and describing its properties (identifying); identifying similarities and differences between thisconcept and others (discriminating); identifying general properties of the concept in particular cases of it(generalizing); and perceiving a unifying principle (synthesizing, defining) (Sierpinska, 1994).The following examples illustrate these practices Work on some of the activities on Handout 8. Here the objects are geometrical shapes. Theobjects could equally well be equations, words, numbers What kinds of 'objects' do you ask students to classify and define in your classroom? Try to develop an activity using one of these types for use in your own classroom. Try todevise examples that force students to observe the properties of objects carefully, and that willcreate discussion about definitions.Try out your activity and report back on it in a later session.The types of activity shown here may be extended to almost any context. The objects being described,defined and classified could be numerical, geometric or algebraic.Similarities and differencesStudents may, for example, decide that the square is the odd one out because it has a different perimeter tothe other shapes (which both have the same perimeter); the rectangle is the odd one out because it has adifferent area to the others and so on. Properties considered may include area, perimeter, symmetry, angle,convexity etc. Participants should try to devise their own examples.Properties and definitionsNone of the properties by themselves defines the square. It is interesting to consider what other shapes areincluded if just one property is taken. For example, when the property is ‘Two equal diagonals’ then allrectangles and isosceles trapezoids are included - but is that all the cases?Taken two at a time, then results are not so obvious. For example, ‘four equal sides’ and ‘four right angles’defines a square, but ‘diagonals meet at right angles’ and ‘four equal sides’ does not (what else could thisbe?).Creating and testing definitionsParticipants usually write a rather vague definition of ‘polygon’ to begin with, such as: "A shape withstraight edges." They then see that this is inadequate for the given examples. This causes them to redefinemore rigorously, like "a plane figure that is bounded by a closed path or circuit, composed of a finitesequence of straight line segments." Defining is difficult, and students should realize that there arecompeting definitions for the same idea (such as ‘dimension’, for example).Classifying using two-way tablesTwo-way tables are not the only representation that may be used, of course, and participants may suggestothers. Venn diagrams and tree diagrams are just two examples.PD Module GuideConcept Development Lessons8
Handout 8: Classifying mathematical objects7 Classifying mathematical objectsSimilarities and differencesShow students three objects."Which is the odd one out?""Describe properties that two share that the third does not.""Now choose a different object from the three and justify itas the odd one out."(a)(b)(c)Properties and definitionsShow students an object."Look at this object and write down all its properties.""Does any single property constitute a definition of theobject? If not, what o
PD Module Guide Concept Development Lessons 2 Activity A: Using the assessment tasks Time needed: 30 minutes. Each Formative Assessment Lesson is preceded by an introductory assessment task. The purpose of this is to discover the in
TOPIC 12 Understand Fractions as Numbers 8 LESSONS 13 DAYS TOPIC 13 Fraction Equivalence and Comparison 8 LESSONS 12 DAYS TOPIC 14 Solve Time, Capacity, and Mass Problems 9 LESSONS 11 DAYS TOPIC 15 Attributes of Two-Dimensional Shapes* 5 LESSONS 9 DAYS TOPIC 16 Solve Perimeter Problems 6 LESSONS 8 DAYS Step Up Lessons 10 LESSONS 10 DAYS TOTAL .
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a concept development method while concept development is a critical element in theory development. Concept analysis for them is "a process of determining the likeness and unlikeness between concepts" and its "basic purpose is to distinguish between the defining attributes of a concept and its irrelevant attributes" (1994:38).
The potential of formative assessment to improve learning. Briefly mention the research evidence that sets out the case for formative assessment. This is summarized by Black and Wiliam in several accessible publications for teachers (see p. 2). These researchers set out to find out whether or not improving
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